CORC Report TR-2003-03 Concurrent Flows in O∗(1 ) iterations

We adapt a method proposed by Nesterov [N03] to obtain an algorithm that computes -optimal solutions to packing problems by solving O∗( −1 √ Kn) separable convex quadratic programs, where K is the maximum number of nonzeros per row and n is the number of variables. We also show one can approximate the solution of the quadratic program to any degree of accuracy by the solution of an appropriately defined piecewise-linear program. For the special case of the maximum concurrent flow problem with rational capacities and demands we obtain an algorithm that computes an -optimal flow by solving O∗( −1K3/2M √ N (log 1 +LU +LD)) shortest path problems, where K is the number of commodities,M is the number of arcs, N is the number of nodes, and LU and LD are, respectively, the number of bits needed to store the capacities and demands. In contrast, previous algorithms required Ω( −2) iterations.